On the Well-Posedness of the Kirchhoff String

Abstract

Abstract. Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equationwhere Ω is an open subset of R n and m is a positive function of one real variable which is continuously differentiable. We prove the well-posedness in the Hadamard sense (existence, uniqueness and continuous dependence of the local solution upon the initial data) in Sobolev spaces of low order.

Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions

“…From (a 4 ) for sufficiently large t >0 we haveâ(t) 3 ) and (g 3 ) are satisfied, then (P) has a sequence of solutions {±u n } such that E(±u n ) +∞ as n ∞.…”

“…Various equations of Kirchhoff type have been studied by many authors, especially after the work of Lions [2], where a functional analysis framework for the problem was proposed; see e.g. [3][4][5][6] for some interesting results and further references. In the following, a key work on nonlocal elliptic problems is the article by Chipot and Rodrigues [7].…”

Existence and multiplicity of positive solutions for a class of p ( x )-Kirchhoff type equations

“…Equation (1.3) received much attention only after Lions [18] proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [19][20][21][22]. Moreover, nonlocal boundary value problems like (1.3) can be used for modeling several physical and biological systems where u describes a process which depends on the average of itself, such as the population density [23][24][25][26].…”

Sign-changing multi-bump solutions for Kirchhoff-type equations in <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^3$\end{document}</tex-math></inline-formula>

“…problem (3) began to call attention of several researchers, see [4,10,13] and the references therein. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations.…”

“…We have to point out that such nonlocal problems also appear in other fields as biological systems, where u describes a process which depends on the average of itself, for example, population density. For more mathematical and physical background of the problem (2), we refer the reader to the papers [1,4,10,21,22,25,26] and the references therein.…”